Stacking - Revision history

Godtone: "or" makes it sound like it's a separate thing, and describing it as a rank 2 temperament is contentious, unconventional and confusing at best, especially when explaining a concept as simple as stacking

2025-03-23T04:27:56Z

"or" makes it sound like it's a separate thing, and describing it as a rank 2 temperament is contentious, unconventional and confusing at best, *especially* when explaining a concept as simple as stacking

← Older revision		Revision as of 04:27, 23 March 2025
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	In tuning theory, '''stacking''' is multiplying or dividing the pitch [[ratio]]s corresponding to the intervals, or adding or subtracting their [[cent]] values. Depending on context, [[octave reduction]] (or the analogue for other [[equave]]s) is sometimes assumed.		In tuning theory, '''stacking''' is multiplying or dividing the pitch [[ratio]]s corresponding to the intervals, or adding or subtracting their [[cent]] values. Depending on context, [[octave reduction]] (or the analogue for other [[equave]]s) is sometimes assumed.

	A simple example of stacking can be seen in [[Pythagorean]] tuning~~, or~~ [[3-limit]] [[JI]]~~, a [[rank-2 temperament]] that [[tempering out\|tempers out]] no [[comma]]s~~, which is generated by stacking (multiplying or dividing by) the primes 3 and 2. For example, the Pythagorean comma [[531441/524288]] can be produced by multiplying by 3 twelve times, and then dividing by 2 nineteen times. Its descending counterpart, 524288/531441, can be produced by multiplying by 2 nineteen times, then dividing by 3 twelve times. In terms of cent values, this corresponds to adding or subtracting steps of 1200 cents or {{nowrap\| log<sub>2</sub>(3) ⋅ 1200 ≈ 1901.955 }} cents.		A simple example of stacking can be seen in [[Pythagorean]] tuning a.k.a. [[3-limit]] [[JI]], which is generated by stacking (multiplying or dividing by) the primes 3 and 2. For example, the Pythagorean comma [[531441/524288]] can be produced by multiplying by 3 twelve times, and then dividing by 2 nineteen times. Its descending counterpart, 524288/531441, can be produced by multiplying by 2 nineteen times, then dividing by 3 twelve times. In terms of cent values, this corresponds to adding or subtracting steps of 1200 cents or {{nowrap\| log<sub>2</sub>(3) ⋅ 1200 ≈ 1901.955 }} cents.

	Stacking is used to explain [[regular temperament]]s, which are often described in terms of stacking multiple instances of a single interval to produce another interval, and commas, which are describable as the difference between a stack of one interval and a stack of a different interval. In a [[rank]]-''n'' [[temperament]], there are ''n'' [[generator]]s which can be stacked to produce any interval in the group.		Stacking is used to explain [[regular temperament]]s, which are often described in terms of stacking multiple instances of a single interval to produce another interval, and commas, which are describable as the difference between a stack of one interval and a stack of a different interval. In a [[rank]]-''n'' [[temperament]], there are ''n'' [[generator]]s which can be stacked to produce any interval in the group.

ArrowHead294 at 12:51, 24 February 2025

2025-02-24T12:51:37Z

← Older revision		Revision as of 12:51, 24 February 2025
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	In tuning theory, '''stacking''' is multiplying or dividing the pitch [[ratio]]s corresponding to the intervals, or adding or subtracting their [[cent]] values. Depending on context, [[octave reduction]] (or the analogue for other [[equave]]s) is sometimes assumed.		In tuning theory, '''stacking''' is multiplying or dividing the pitch [[ratio]]s corresponding to the intervals, or adding or subtracting their [[cent]] values. Depending on context, [[octave reduction]] (or the analogue for other [[equave]]s) is sometimes assumed.

	A simple example of stacking can be seen in [[Pythagorean]] tuning, or [[3-limit]] [[JI]], a [[rank-2 temperament]] that [[tempering out\|tempers out]] no [[comma]]s, which is generated by stacking (multiplying or dividing by) the primes 3 and 2. For example, the Pythagorean comma [[531441/524288]] can be produced by multiplying by 3 twelve times, and then dividing by 2 nineteen times. Its descending counterpart, 524288/531441, can be produced by multiplying by 2 nineteen times, then dividing by 3 twelve times. In terms of cent values, this corresponds to adding or subtracting steps of 1200 cents or {{nowrap\| log<sub>2</sub>(3)~~⋅1200~~ ≈ 1901.955 }} cents.		A simple example of stacking can be seen in [[Pythagorean]] tuning, or [[3-limit]] [[JI]], a [[rank-2 temperament]] that [[tempering out\|tempers out]] no [[comma]]s, which is generated by stacking (multiplying or dividing by) the primes 3 and 2. For example, the Pythagorean comma [[531441/524288]] can be produced by multiplying by 3 twelve times, and then dividing by 2 nineteen times. Its descending counterpart, 524288/531441, can be produced by multiplying by 2 nineteen times, then dividing by 3 twelve times. In terms of cent values, this corresponds to adding or subtracting steps of 1200 cents or {{nowrap\| log<sub>2</sub>(3) ⋅ 1200 ≈ 1901.955 }} cents.

	Stacking is used to explain [[regular temperament]]s, which are often described in terms of stacking multiple instances of a single interval to produce another interval, and commas, which are describable as the difference between a stack of one interval and a stack of a different interval. In a [[rank]]-''n'' [[temperament]], there are ''n'' [[generator]]s which can be stacked to produce any interval in the group.		Stacking is used to explain [[regular temperament]]s, which are often described in terms of stacking multiple instances of a single interval to produce another interval, and commas, which are describable as the difference between a stack of one interval and a stack of a different interval. In a [[rank]]-''n'' [[temperament]], there are ''n'' [[generator]]s which can be stacked to produce any interval in the group.
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	a) It follows the group theory axioms ("group"):		a) It follows the group theory axioms ("group"):
	* Stacking a [[unison]] on another interval produces no change; in other words, the unison is the identity element. For example, {{nowrap\| 5/4 ⋅ 1 = 5/4 }}. This corresponds to the number 1 being the multiplicative identity.		* Stacking a [[unison]] on another interval produces no change; in other words, the unison is the identity element. For example, {{nowrap\| 5/4 ⋅ 1 {{=}} 5/4 }}. This corresponds to the number 1 being the multiplicative identity.
	* Stacking intervals is associative; for example {{nowrap\| [[5/4]] ⋅ ([[6/5]] ⋅ [[4/3]]) = (5/4 ⋅ 6/5) ⋅ 4/3 = [[2/1\|2]] }}. This corresponds to the fact that multiplication is associative.		* Stacking intervals is associative; for example {{nowrap\| [[5/4]] ⋅ ([[6/5]] ⋅ [[4/3]]) {{=}} (5/4 ⋅ 6/5) ⋅ 4/3 {{=}} [[2/1\|2]] }}. This corresponds to the fact that multiplication is associative.
	* There are ascending and descending versions of each interval. Stacking the ascending and descending versions of an interval produces the unison. For example, {{nowrap\| 5/4 ⋅ 4/5 = 1 }}. This corresponds to the descending interval being the multiplicative inverse of the ascending interval.		* There are ascending and descending versions of each interval. Stacking the ascending and descending versions of an interval produces the unison. For example, {{nowrap\| 5/4 ⋅ 4/5 {{=}} 1 }}. This corresponds to the descending interval being the multiplicative inverse of the ascending interval.
	* Stacking intervals in such a group produces another interval in the group. For example, 5/4 and 6/5 are in the group of just intervals, and thus it follows that [[3/2]] is as well.		* Stacking intervals in such a group produces another interval in the group. For example, 5/4 and 6/5 are in the group of just intervals, and thus it follows that [[3/2]] is as well.

	b) Stacking intervals is commutative ("abelian"):		b) Stacking intervals is commutative ("abelian"):
	* For example, {{nowrap\| 5/4 ⋅ 6/5 = 6/5 ⋅ 5/4 = 3/2 }}. This corresponds to the fact that multiplication of real numbers is commutative.		* For example, {{nowrap\| 5/4 ⋅ 6/5 {{=}} 6/5 ⋅ 5/4 {{=}} 3/2 }}. This corresponds to the fact that multiplication of real numbers is commutative.

	c) Nontrivial products of stacks of generators do not ever produce the identity ("free").		c) Nontrivial products of stacks of generators do not ever produce the identity ("free").

FloraC: Cleanup and +categories

2025-02-21T11:11:52Z

Cleanup and +categories

← Older revision		Revision as of 11:11, 21 February 2025
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	In ~~the context of~~ tuning theory, '''stacking''' ~~refers to~~ multiplying or dividing the pitch [[ratio]]s corresponding to the intervals, or adding or subtracting their [[cent]] values. Depending on context, [[octave reduction]] (or the analogue for other [[equave]]s) is sometimes assumed~~. In a [[rank]]-''n'' [[temperament]], there are ''n'' [[generator]]s which can be stacked to produce any interval in the group~~.		In tuning theory, '''stacking''' is multiplying or dividing the pitch [[ratio]]s corresponding to the intervals, or adding or subtracting their [[cent]] values. Depending on context, [[octave reduction]] (or the analogue for other [[equave]]s) is sometimes assumed.

	A simple example of stacking can be seen in [[Pythagorean]] tuning, or [[3-limit]] [[JI]], a [[rank-2 temperament]] that [[~~Tempering~~ out\|tempers out]] no [[comma]]s, which is generated by stacking (multiplying or dividing by) the primes 3 and 2. For example, the ~~interval~~ [[531441/524288]] can be produced by multiplying by 3 twelve times, and then dividing by 2 nineteen times. Its descending counterpart, 524288/531441, can be produced by multiplying by 2 nineteen times, then dividing by 3 twelve times. In terms of cent values, this corresponds to adding or subtracting steps of 1200 cents or ~~log2~~(3)*1200 ≈ 1901.955 cents.		A simple example of stacking can be seen in [[Pythagorean]] tuning, or [[3-limit]] [[JI]], a [[rank-2 temperament]] that [[tempering out\|tempers out]] no [[comma]]s, which is generated by stacking (multiplying or dividing by) the primes 3 and 2. For example, the Pythagorean comma [[531441/524288]] can be produced by multiplying by 3 twelve times, and then dividing by 2 nineteen times. Its descending counterpart, 524288/531441, can be produced by multiplying by 2 nineteen times, then dividing by 3 twelve times. In terms of cent values, this corresponds to adding or subtracting steps of 1200 cents or {{nowrap\| log<sub>2</sub>(3)⋅1200 ≈ 1901.955 }} cents.

	Stacking is used to explain [[regular temperament]]s, which are often described in terms of stacking multiple instances of a single interval to produce another interval, and commas, which are describable as the difference between a stack of one interval and a stack of a different interval.		Stacking is used to explain [[regular temperament]]s, which are often described in terms of stacking multiple instances of a single interval to produce another interval, and commas, which are describable as the difference between a stack of one interval and a stack of a different interval. In a [[rank]]-''n'' [[temperament]], there are ''n'' [[generator]]s which can be stacked to produce any interval in the group.

	== Mathematical definition ==		== Mathematical definition ==
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	Stacking is the group operation on a free abelian group of musical intervals.		Stacking is the group operation on a free abelian group of musical intervals.

	In the context of group theory, any space of [[~~Interval\|intervals~~]] created by multiplicatively stacking arbitrarily many (or negatively many) of a given set of [[~~Generator\|generators~~]] is considered a '''free abelian group''' under stacking. Where the set of generators is finite, it is called a '''finitely generated free abelian group'''.		In the context of group theory, any space of [[interval]]s created by multiplicatively stacking arbitrarily many (or negatively many) of a given set of [[generator]]s is considered a '''free abelian group''' under stacking. Where the set of generators is finite, it is called a '''finitely generated free abelian group'''.

	The more explicit definition for this follows.		The more explicit definition for this follows.

	''Note: Examples are provided with the group of [[~~Just~~ intonation\|just intervals]], corresponding to the infinite set of primes as generators, and to positive rationals under multiplication. However, the following properties can be shown to apply in other cases, such as tunings of [[~~Regular~~ temperament~~\|regular temperaments~~]], or any specific [[subgroup]] of JI, including [[~~Prime~~ limit~~\|prime limits~~]]. An intuitive way to confirm this is to see that any set of generators can be seen as a tempering of some just intonation subgroup, if a very inaccurate one.''		'''Note:''' Examples are provided with the group of [[just intonation\|just intervals]], corresponding to the infinite set of primes as generators, and to positive rationals under multiplication. However, the following properties can be shown to apply in other cases, such as tunings of [[regular temperament]]s, or any specific [[subgroup]] of JI, including [[prime limit]]s. An intuitive way to confirm this is to see that any set of generators can be seen as a tempering of some just intonation subgroup, if a very inaccurate one.

	a) it follows the group theory axioms ("group"):		a) It follows the group theory axioms ("group"):
			* Stacking a [[unison]] on another interval produces no change; in other words, the unison is the identity element. For example, {{nowrap\| 5/4 ⋅ 1 = 5/4 }}. This corresponds to the number 1 being the multiplicative identity.
	* Stacking a [[unison]] on another interval produces no change; in other words, the unison is the identity element. For example, 5/4 * 1 = 5/4. This corresponds to the number 1 being the multiplicative identity.		* Stacking intervals is associative; for example {{nowrap\| [[5/4]] ⋅ ([[6/5]] ⋅ [[4/3]]) = (5/4 ⋅ 6/5) ⋅ 4/3 = [[2/1\|2]] }}. This corresponds to the fact that multiplication is associative.
	* Stacking intervals is associative; for example [[5/4]] * ([[6/5]] * [[4/3]]) = (5/4 * 6/5) * 4/3 = [[2/1\|2]]. This corresponds to the fact that multiplication is associative.		* There are ascending and descending versions of each interval. Stacking the ascending and descending versions of an interval produces the unison. For example, {{nowrap\| 5/4 ⋅ 4/5 = 1 }}. This corresponds to the descending interval being the multiplicative inverse of the ascending interval.
	* There are ascending and descending versions of each interval. Stacking the ascending and descending versions of an interval produces the ~~[[1/1\|~~unison]]. For example, 5/4 * 4/5 = 1. This corresponds to the descending interval being the multiplicative inverse of the ascending interval.		* Stacking intervals in such a group produces another interval in the group. For example, 5/4 and 6/5 are in the group of just intervals, and thus it follows that [[3/2]] is as well.
	* Stacking intervals in such a group produces another interval in the group. For example, 5/4 and 6/5 are in the group of ~~[[Just intonation\|~~just intervals]], and thus it follows that [[3/2]] is as well.

	b) Stacking intervals is commutative ("abelian"):		b) Stacking intervals is commutative ("abelian"):
			* For example, {{nowrap\| 5/4 ⋅ 6/5 = 6/5 ⋅ 5/4 = 3/2 }}. This corresponds to the fact that multiplication of real numbers is commutative.

	* For example, 5/4 * 6/5 = 6/5 * 5/4 = 3/2. This corresponds to the fact that multiplication of real numbers is commutative.		c) Nontrivial products of stacks of generators do not ever produce the identity ("free").

	c) Nontrivial products of stacks of generators ~~don't~~ ever produce the identity ("free").

	* In the case of JI, this corresponds to the fundamental theorem of arithmetic, because every interval has a unique prime factorization.		* In the case of JI, this corresponds to the fundamental theorem of arithmetic, because every interval has a unique prime factorization.
	* In the case of tunings such as [[~~Equal-step tuning\|~~equal ~~temperaments~~]], there is only one generator.		* In the case of tunings such as [[equal temperament]]s, there is only one generator.
	* The number of generators is the [[rank]] of the temperament, so that equal temperaments are rank-1, temperaments with a generator and period are rank-2, and so on.		* The number of generators is the [[rank]] of the temperament, so that equal temperaments are rank-1, temperaments with a generator and period are rank-2, and so on.

			[[Category:Terms]]
			[[Category:Math]]
	[[Category:Method]]		[[Category:Method]]

VectorGraphics at 03:39, 21 February 2025

2025-02-21T03:39:30Z

← Older revision		Revision as of 03:39, 21 February 2025
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	== Mathematical definition ==		== Mathematical definition ==
			{{Wikipedia\|Free abelian group}}
	Stacking is the group operation on a free abelian group of musical intervals.		Stacking is the group operation on a free abelian group of musical intervals.

	In the context of group theory, any space of [[Interval\|intervals]] created by multiplicatively stacking arbitrarily many (or negatively many) of a given set of [[Generator\|generators]] is considered a '''free abelian group''' under stacking.		In the context of group theory, any space of [[Interval\|intervals]] created by multiplicatively stacking arbitrarily many (or negatively many) of a given set of [[Generator\|generators]] is considered a '''free abelian group''' under stacking. Where the set of generators is finite, it is called a '''finitely generated free abelian group'''.

	The more explicit definition for this follows.		The more explicit definition for this follows.
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	* In the case of tunings such as [[Equal-step tuning\|equal temperaments]], there is only one generator.		* In the case of tunings such as [[Equal-step tuning\|equal temperaments]], there is only one generator.
	* The number of generators is the [[rank]] of the temperament, so that equal temperaments are rank-1, temperaments with a generator and period are rank-2, and so on.		* The number of generators is the [[rank]] of the temperament, so that equal temperaments are rank-1, temperaments with a generator and period are rank-2, and so on.

	[[Category:Method]]		[[Category:Method]]

VectorGraphics at 03:35, 21 February 2025

2025-02-21T03:35:39Z

@@ Line 1: / Line 1: @@
-In the context of tuning theory, '''stacking''' is the group operation of a [[free abelian group]] of [[interval]]s. It corresponds to multiplying or dividing the pitch [[ratio]]s corresponding to the intervals, or adding or subtracting their [[cent]] values. Depending on context, [[octave reduction]] (or the analogue for other [[equave]]s) is sometimes assumed. In a [[rank]]-''n'' [[temperament]], there are ''n'' [[generator]]s which can be stacked to produce any interval in the group.
+In the context of tuning theory, '''stacking''' refers to multiplying or dividing the pitch [[ratio]]s corresponding to the intervals, or adding or subtracting their [[cent]] values. Depending on context, [[octave reduction]] (or the analogue for other [[equave]]s) is sometimes assumed. In a [[rank]]-''n'' [[temperament]], there are ''n'' [[generator]]s which can be stacked to produce any interval in the group.
 A simple example of stacking can be seen in [[Pythagorean]] tuning, or [[3-limit]] [[JI]], a [[rank-2 temperament]] that [[Tempering out|tempers out]] no [[comma]]s, which is generated by stacking (multiplying or dividing by) the primes 3 and 2. For example, the interval [[531441/524288]] can be produced by multiplying by 3 twelve times, and then dividing by 2 nineteen times. Its descending counterpart, 524288/531441, can be produced by multiplying by 2 nineteen times, then dividing by 3 twelve times. In terms of cent values, this corresponds to adding or subtracting steps of 1200 cents or log2(3)*1200 ≈ 1901.955 cents.
 Stacking is used to explain [[regular temperament]]s, which are often described in terms of stacking multiple instances of a single interval to produce another interval, and commas, which are describable as the difference between a stack of one interval and a stack of a different interval.
 [[Category:Method]]

Lériendil: added links and fixed markup

2025-02-21T02:27:18Z

added links and fixed markup

← Older revision		Revision as of 02:27, 21 February 2025
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	In the context of tuning theory, '''stacking''' is the group operation of a [[free abelian group]] of [[interval]]~~<nowiki/>~~s. It corresponds to multiplying or dividing the pitch [[ratio]]~~<nowiki/>~~s corresponding to the intervals, or adding or subtracting their [[cent]] values. In a rank-n [[temperament]], there are n [[~~Generator\|generators~~]] which can be stacked to produce any interval in the group.		In the context of tuning theory, '''stacking''' is the group operation of a [[free abelian group]] of [[interval]]s. It corresponds to multiplying or dividing the pitch [[ratio]]s corresponding to the intervals, or adding or subtracting their [[cent]] values. Depending on context, [[octave reduction]] (or the analogue for other [[equave]]s) is sometimes assumed. In a [[rank]]-''n'' [[temperament]], there are ''n'' [[generator]]s which can be stacked to produce any interval in the group.

	A simple example of stacking can be seen in [[Pythagorean]] tuning, or 3-limit JI, a [[rank-2 temperament]] that [[Tempering out\|tempers out]] no c[[~~Comma\|ommas~~]], which is generated by stacking (multiplying or dividing by) the primes 3 and 2. For example, the interval [[531441/524288]] can be produced by multiplying by 3 twelve times, and then dividing by 2 nineteen times. Its descending counterpart, 524288/531441, can be produced by multiplying by 2 nineteen times, then dividing by 3 twelve times. In terms of cent values, this corresponds to adding or subtracting steps of 1200 cents or log2(3)*1200 ≈ 1901.955 cents.		A simple example of stacking can be seen in [[Pythagorean]] tuning, or [[3-limit]] [[JI]], a [[rank-2 temperament]] that [[Tempering out\|tempers out]] no [[comma]]s, which is generated by stacking (multiplying or dividing by) the primes 3 and 2. For example, the interval [[531441/524288]] can be produced by multiplying by 3 twelve times, and then dividing by 2 nineteen times. Its descending counterpart, 524288/531441, can be produced by multiplying by 2 nineteen times, then dividing by 3 twelve times. In terms of cent values, this corresponds to adding or subtracting steps of 1200 cents or log2(3)*1200 ≈ 1901.955 cents.

	Stacking is used to explain [[~~Regular~~ temperament~~\|regular temperaments~~]], which are often described in terms of stacking an interval to produce another interval.		Stacking is used to explain [[regular temperament]]s, which are often described in terms of stacking multiple instances of a single interval to produce another interval, and commas, which are describable as the difference between a stack of one interval and a stack of a different interval.

			[[Category:Method]]

VectorGraphics: Created page with "In the context of tuning theory, '''stacking''' is the group operation of a free abelian group of intervals. It corresponds to multiplying or dividing the pit..."

2025-02-21T02:22:46Z

Created page with "In the context of tuning theory, '''stacking''' is the group operation of a free abelian group of interval<nowiki/>s. It corresponds to multiplying or dividing the pit..."

New page

In the context of tuning theory, '''stacking''' is the group operation of a [[free abelian group]] of [[interval]]<nowiki/>s. It corresponds to multiplying or dividing the pitch [[ratio]]<nowiki/>s corresponding to the intervals, or adding or subtracting their [[cent]] values. In a rank-n [[temperament]], there are n [[Generator|generators]] which can be stacked to produce any interval in the group.

A simple example of stacking can be seen in [[Pythagorean]] tuning, or 3-limit JI, a [[rank-2 temperament]] that [[Tempering out|tempers out]] no c[[Comma|ommas]], which is generated by stacking (multiplying or dividing by) the primes 3 and 2. For example, the interval [[531441/524288]] can be produced by multiplying by 3 twelve times, and then dividing by 2 nineteen times. Its descending counterpart, 524288/531441, can be produced by multiplying by 2 nineteen times, then dividing by 3 twelve times. In terms of cent values, this corresponds to adding or subtracting steps of 1200 cents or log2(3)*1200 ≈ 1901.955 cents.

Stacking is used to explain [[Regular temperament|regular temperaments]], which are often described in terms of stacking an interval to produce another interval.